Measurable Function

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measurable function

[′mezh·rə·bəl ′fəŋk·shən]
A real valued function ƒ defined on a measurable space X, where for every real number a all those points x in X for which ƒ(x) ≥ a form a measurable set.
A function on a measurable space to a measurable space such that the inverse image of a measurable set is a measurable set.

Measurable Function


(in the original meaning), a function f(x) that has the property that for any t the set Et of points x, for which each f(x) ≤ t, is Lebesgue measurable. This definition of a measurable function was given by the French mathematician H. Lebesgue. The sum, difference, product, and quotient of two measurable functions, as well as the limit of a sequence of measurable functions, are in turn measurable functions. Thus, the basic operations of algebra and analysis do not go beyond the framework of the set of measurable functions. Russian and Soviet mathematicians have made a major contribution to the study of measurable functions (D. F. Egorov, N.N. Luzin, and their students). Luzin proved that a function is measurable if and only if it can be made continuous after its values are varied in a set of as small as desired measure. This is the so-called C-property of measurable functions.

In the abstract theory of measure, the function f(x) is said to be a measurable function with respect to some measure μ. if the set Et is found in the domain of definition of the measure μ. In modern probability theory measurable functions are called random variables.

References in periodicals archive ?
A Lebesgue measurable function A is a (q, N)-atom for [H.
If 1/q - 1/r < [mu] [less than or equal to] 1/q, then for any Lebesgue measurable function a satisfying
For any f 2 Hp (R) with suppf [subset or equal to] [0, [infinity]), we have a family of Lebesgue measurable functions {[a.
In the general case of any Lebesgue measurable function f with bounded variation M > 0 the inequality of Lemma 3.
s](T,Z) the space of all (equivalence classes of) strongly Lebesgue measurable functions w : T [right arrow] Z such that t [right arrow] [[parallel]w(t)[parallel].
M,[omega]] (T) the class of Lebesgue measurable functions f : T [right arrow] C satisfying the condition